Surds

Tags

math

Refers to numbers containing square root symbols, such as \( \sqrt{12}, \sqrt{3}\sqrt{5}, 5 + \sqrt{78} \) etc.

Surds can be simplified, split up and otherwise operated on like regular numbers. For example \( \sqrt{5} = \sqrt{5} \sqrt{1} \), or more practically:

\begin{align*} \sqrt{48} &= \sqrt{48} \\

        &= \sqrt{16} \sqrt{3} \\\\
        &= 4 \sqrt{3}

\end{align*}

In many cases you may end up with a fraction that contains a surd on its denominator, in this situation we'd often like to rationalise it to a regular number. This can be done by multiplying both the numerator and the denominator by the by the surd, cancelling it out in the denominator and making the numerator a surd.

\begin{align*} \frac{5}{\sqrt{3}} &= \frac{5}{\sqrt{3}} \frac{\sqrt{3}}{\sqrt{3}} \\

                 &= \frac{5 \sqrt{3}}{\sqrt{3} \sqrt{3}} \\\\
                 &= \frac{5 \sqrt{3}}{3}

\end{align*}

For denominators containing a summation or more complex form you can rationalise using a form that cancels out the surd in the denominator.

\begin{align*} \frac{7}{4 + \sqrt{2}} &= \frac{7}{4 + \sqrt{2}} \frac{4 - \sqrt{2}}{4 - \sqrt{2}} \\

                     &= \frac{7 \times 4 + 7 \times \sqrt{2}}{(4 + \sqrt{2})(4 + \sqrt{2})} \\\\
                     &= \frac{28 + 7 \sqrt{2}}{16 + 4 \sqrt{2} - 4 \sqrt{2} + 2} \\\\
                     &= \frac{28 + 7 \sqrt{2}}{18}

\end{align*}